Independent events A and B with P(A) = 1/2 and P(B) = 1/3 give P(A ∩ B) = 1/6, and conditional probability is P(A ∩ B) ÷ P(B), so this Class 12 sheet opens with the multiplication rule and its conditional ratio. A 1/16 answer follows for a coin problem.
Key Takeaways
P(A ∩ B) = 1/6 for independent events with P(A) = 1/2 and P(B) = 1/3 (Q1).
P(A|B) = P(A ∩ B) ÷ P(B) (Q2).
Q4 records 1/16 among the answers.
Section A: Multiple Choice Questions (1 Mark Each)
Choose the correct option for each question.
1.If P(A) = 1/2, P(B) = 1/3 and the events A and B are independent, then P(A ∩ B) = ?
(a) 1/5(b) 1/6(c) 5/6(d) 1/2
2.The conditional probability P(A|B) is defined as:
(a) P(A ∩ B) / P(B)(b) P(A ∪ B) / P(B)(c) P(B) / P(A)(d) P(A) × P(B)
3.A random variable X takes values 0 and 1 with P(X = 0) = P(X = 1) = 1/2. The expected value E(X) is:
(a) 0(b) 1/2(c) 1(d) 1/4
4.In a binomial distribution with n = 4 and p = 1/2, the value of P(X = 0) is:
(a) 1/16(b) 1/8(c) 1/4(d) 0
5.Two events A and B are independent if:
(a) P(A ∩ B) = P(A) × P(B)(b) P(A ∪ B) = P(A) + P(B)(c) A ∩ B = ∅(d) P(A) = P(B)
6.If P(A) = 0.3, P(B) = 0.4 and A and B are mutually exclusive, then P(A ∪ B) = ?
(a) 0.7(b) 0.12(c) 0.58(d) 0.1
Section B: Short Answer Type Questions (2 Marks Each)
Show all steps clearly.
7.Write the formula for the conditional probability P(A|B).
8.If P(A) = 0.5, P(B) = 0.4 and P(A ∩ B) = 0.2, find P(A|B).
9.Define independent events.
10.A fair die is rolled once. What is the probability of getting a prime number?
11.State the formula for P(X = r) in a binomial distribution with n trials and probability of success p.
12.For a binomial variable with n = 3 and p = 1/3, find P(X = 3).
13.If E(X) = 2 and Var(X) = 1, find E(X²).
14.Two dice are thrown once. Find the probability of getting a total of 7.
Section C: Numericals & Word Problems (3 Marks Each)
Apply the concepts to solve the problems. Show all working.
15.A bag contains 3 red and 5 black balls. Two balls are drawn one after another without replacement. Find the probability that both balls are red.
16.A fair die is thrown twice. Find the probability that both throws show the same number.
17.In a class, 60% of the students passed Mathematics and 45% passed both Mathematics and Science. A student is chosen at random and is found to have passed Mathematics. Find the probability that the student also passed Science.
18.A fair coin is tossed 4 times. Find the probability of getting exactly 2 heads.
19.The probability that a student passes an examination is 2/3. If the student appears in 3 independent examinations, find the probability that the student passes all 3 examinations.
20.A box contains 4 defective and 6 non-defective bulbs. Two bulbs are drawn one after another without replacement. Find the probability that both bulbs drawn are non-defective.
Answer Key
1. b) 1/6
2. a) P(A ∩ B) / P(B)
3. b) 1/2
4. a) 1/16
5. a) P(A ∩ B) = P(A) × P(B)
6. a) 0.7
7. Refer to solution guide.
8. Refer to solution guide.
9. Refer to solution guide.
10. Refer to solution guide.
11. Refer to solution guide.
12. Refer to solution guide.
13. Refer to solution guide.
14. Refer to solution guide.
15. Refer to solution guide.
16. Refer to solution guide.
17. Refer to solution guide.
18. Refer to solution guide.
19. Refer to solution guide.
20. Refer to solution guide.
End of Worksheet
People Also Ask
What is P(A ∩ B) when P(A) = 1/2 and P(B) = 1/3?
1/6, option (b) in this worksheet (Q1).
How is P(A|B) defined?
P(A ∩ B) ÷ P(B), option (a) in this worksheet (Q2).
What is P(A ∩ B) when A and B are independent?
P(A) × P(B), option (a) in this worksheet (Q5).
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